19N.1.AHL.TZ0.H_3

pestleMathematicsAAHLPaper 119N· ahl-1-16-solution-of-systems-of-linear-equationssource ↗

Three planes have equations:

$2x - y + z = 5$

$x + 3y - z = 4$ , where $a{\text{, }}b \in \mathbb{R}$ .

$3x - 5y + az = b$

Find the set of values of $a$ and $b$ such that the three planes have no points of intersection.

Markscheme / solution

attempt to eliminate a variable (or attempt to find det $A$ ) M1

$\left( {\left. {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ 1&3&{ - 1} \\ 3&{ - 5}&a \end{array}\,} \right|\begin{array}{*{20}{c}} 5 \\ 4 \\ b \end{array}} \right) \to \left( {\left. {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ 0&7&{ - 3} \\ 0&{ - 14}&{a + 3} \end{array}\,} \right|\begin{array}{*{20}{c}} 5 \\ 3 \\ {b - 12} \end{array}} \right)$ (or det $A = 14\left( {a - 3} \right)$ )

(or two correct equations in two variables) A1

$\to \left( {\left. {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ 0&7&{ - 3} \\ 0&{ 0}&{a - 3} \end{array}\,} \right|\begin{array}{*{20}{c}} 5 \\ 3 \\ {b - 6} \end{array}} \right)$ (or solving det $A = 0$ )

(or attempting to reduce to one variable, e.g. $\left( {a - 3} \right)z = b - 6$ ) M1

$a = 3{\text{, }}b \ne 6$ A1A1

[5 marks]

Examiners’ report

[N/A]